• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1463-1474
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• 2019

Let C be a nonsingular projective curve of genus ${\geq}2$ over an algebraically closed field of characteristic 0. For a point P in C, the Weierstrass semigroup H(P) is defined as the set of non-negative integers n for which there exists a rational function f on C such that the order of the pole of f at P is equal to n, and f is regular away from P. A point P in C is referred to as a weak Galois-Weierstrass point if P is a Weierstrass point and there exists a Galois morphism ${\varphi}:C{\rightarrow}{\mathbb{p}}^1$ such that P is a total ramification point of ${\varphi}$. In this paper, we investigate the number of weak Galois-Weierstrass points of which the Weierstrass semigroups are generated by two positive integers.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.57-65
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• 1998

In this paper, we consider properties of Dedekind eta-function, modular discrimiant, thata-series and Weierstrass ${\wp}$-function. We prove the integrablities of ${\Delta}({\tau})$ and ${\eta}({\tau})$. Also, we give explicit formulae about ${\Delta}({\tau})$ and ${\tau}(n)$.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.507-517
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• 2004

Let (2,2,2,2) be ramification indices for the Riemann sphere. It is well known that the regular branched covering map corresponding to this, is the Weierstrass P function. Lattes [7] gives a rational function R(z)= ${\frac{z^4+{\frac{1}{2}}g2^{z}^2+{\frac{1}{16}}g{\frac{2}{2}}$ which is chaotic on ${\bar{C}}$ and is induced by the Weierstrass P function and the linear map L(z) = 2z on complex plane C. It is also known that there exist regular branched covering maps from $T^2$ onto ${\bar{C}}$ if and only if the ramification indices are (2,2,2,2), (2,4,4), (2,3,6) and (3,3,3), by the Riemann-Hurwitz formula. In this paper we will construct regular branched covering maps corresponding to the ramification indices (2,4,4), (2,3,6) and (3,3,3), as well as chaotic maps induced by these regular branched covering maps.

• Honam Mathematical Journal
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• v.4 no.1
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• pp.1-12
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• 1982

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.1057-1072
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• 2023

Let 𝜑 : ℝⁿ × [0, ∞) → [0, ∞) be a growth function and H^𝜑(ℝⁿ) the Musielak-Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called 𝜃-summability is considered for multi-dimensional Fourier transforms in H^𝜑(ℝⁿ). Precisely, with some assumptions on 𝜃, the authors first prove that the maximal operator of the 𝜃-means is bounded from H^𝜑(ℝⁿ) to L^𝜑(ℝⁿ). As consequences, some norm and almost everywhere convergence results of the 𝜃-means, which generalizes the well-known Lebesgue's theorem, are then obtained. Finally, the corresponding conclusions of some specific summability methods, such as Bochner-Riesz, Weierstrass and Picard-Bessel summations, are also presented.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.693-704
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• 2012

We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n{\geq}1$, we have $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1;3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.

• Journal for History of Mathematics
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• v.30 no.1
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• pp.31-50
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• 2017

This study analyses on the history of uniform convergence, and discusses its educational implications. First, this study inspects 'overflowing of the Euclidean methodology' which was suggested by Lakatos as a cause of tardy appearance of uniform convergence, and reinterprets that cause in the perspective of 'symbolization'. Second, this study looks into the emergence of uniform convergence of Seidel and Weierstrass in this viewpoint of symbolization. As a result, of analysis, we come to know that the definition of uniform convergence had been changed into the theory of 'domain and graph' from that of 'point and function value' by the location change of the quantifier. As these results, this study puts forward an educational suggestion from an angle of epistemological obstacle, concept definition and concept image.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.675-683
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• 2000

As a by-product of [4], we give algebraic integers of certain values of quotients of Weierstrass $\delta'(\tau),\delta'(\tau)$-functions. We also show that special values of elliptic functions are transcendental numbers.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.759-769
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• 2006

Let L : $C\;\rightarrow\;C$ be a hyperbolic automorphism. Then the hyperbolic toral automorphism $L_A\;T^2\;\rightarrow\;T^2$, induced by L, is a chaotic map ([2] pg.192). We characterize hyperbolic toral automorphisms by proving the converse of the above statement.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.433-442
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• 2010

$\tilde{G}(P,\;Q)$, a new generalization of the set of gap numbers of a pair of points, was described in [1]. Here we study gap numbers of local subring $R\;=\;\cal{K}\;+\;C$ of algebraic function field over a finite field and we give a formula for the number of elements of $\tilde{G}(P,\;Q)$ depending on pure gaps and R.